On Double Set-Function Sugeno Integrals

The Sugeno integral as a typical non-additive integral, is widely used in decision-making, comprehensive evaluation, and classification problems involving uncertainty or fuzziness. The concept of Sugeno integrals originated from Sugeno, and many scholars have effectively promoted its application. To this day, further development of Sugeno integrals remains a significant research direction. This paper aims to further advance the theory of Sugeno integrals by first introducing a new integral, known as the double set-function Sugeno integral (DSSI). This DSSI extends the original Sugeno integral, which was based on a single fuzzy measure, to one that is based on both set-functions and fuzzy measures. The paper also explores the monotonicity of this integral and its Jensen’s inequality, among other properties. Secondly, it discusses the convergence theorems for two types of integral sequences: those involving set-function sequences and those involving function sequences. It derives several convergence theorems, including the monotone convergence theorems, Fatou’s lemmas, and dominated convergence theorems. Finally, the paper examines discrete DSSI, provides its specific representation, and demonstrates through examples that it outperforms classical Sugeno integrals in decision-making problems.